I’ve just done a bit of reading about them, and although they are herbivores, apparently they can be quite viscious. Therefore I would not keep the wombat in the house with the cats, but would build it a wombat house in the yard.

I lurv wombats, and when Down Under was told it was from the aboriginal word, which was closer to ‘whomm-baj’. Certainly some Ozzies pronounce it with the ‘wh’ in the back of their throat.
True story: was on a train with a bunch of grizzled old farmers, telling them early on my first circumnavigation of Australia that I hadn’t yet seen much wildlife. They politely kept their eyes peeled for the foreigner, and suddenly one pointed, saying, ‘There’s a wooly-back wombat! Scads of ’em!’ Of course, it was a herd of sheep 😉
Since have seen lots of the hairy-nosed variety. Love Australians, too!